Local Index Formula on the Equatorial Podleś Sphere

Abstract

We discuss spectral properties of the equatorial Podleś sphere Sq2. As a preparation we also study the ‘degenerate’ (i.e. q=0) case (related to the quantum disk). Over Sq2 we consider two different spectral triples:one related to the Fock representation of the Toeplitz algebra and the isopectral one given in [7]. After the identification of the smooth pre-C*-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the C*-algebra (the nontrivial generator of the K0-group) as well as the pairing with the q-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.

Higson, N.: The residue index theorem. Lecture notes for the 2000 Clay Institute symposium on NCG; The local index formula in noncommutative geometry, Lectures given at the School on Algebraic K-Theory and its applications, Trieste 2002. http://www.math.psu.edu/higson/ResearchPapers.html